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7/26/2019 k&j&cod of LEFM

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3. FRACTURE MECHANICS CONCEPTS

The wellknown three fracture mechanics concepts they are the stress intensity

factor for linear elastic materials crack opening displacement (COD) andJintegral for elasticplastic materials.

3.1 Stress Intensity Factor ()Many years after the Griffith fracture criterion for ideally brittle materials was

established, Irwin suggested a modification that would extend the Griffith theory to

metals exhibiting plastic deformation. He examined the equations that had been

developed for the stresses in the vicinity of a sharp crack in a large plate as illustrated

in Fig.(3.1) .

Fig. (3.1) crack in an infinite plate

The equations for the elastic stress distribution at the crack tip are as follows: = cos

[1 sin sin3 ] 3.1

= cos [1 + sin sin 3 ] 3.2

= cos cos cos3 3.3

And

= cos Plane strain


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Or

= 0 Plane stress

Fig. (3.2) stress distribution at the crack tip

AS should be expected.in the elastic case the stresses are proportional to the external

stress . They vary with the square root of the crack size and they tend to infinity atthe crack tip where r is small. The distribution of the stress as a function of rat =0is illustrated in figure 3.2. For large rthe stress approaches zero, while it shouldgo to.Apparently. eq. (3.1) are valid only for a limited area around the crack tip.Each of the equations represents the first term of a series. In the vicinity of the crack tip

these first terms give a sufficiently accurate description of the crack tip stress fields.

Irwin has developed the stress intensity factor

and defined as

= 3.4Where:

normal stress.ahalf crack length.

Ygeometry factor depends on the crack configuration.

The factor is known as "stress intensity factor" is mean of characterizing the elasticstress distribution near the crack tip but itself has no physical reality. Its depends on theconfiguration of the system and has units of and should not be confused with theelastic stress concentration factor. Where the subscript I stands mode I. Broek[2]quotes interesting statistics, according to which 90% of the engineering problems

involving fracture mechanics are of the Mode I type, another 8% of the combined-mode

type, which, immediately upon initiation of loading, transform into Mode I crack

behavior.


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Fig. (3.3) opening mode

The whole stress field at the crack tip is known when the stress intensity factor is

known.

Crack extension will occur when the stresses and strains at the crack tip reach a critical

value. This means that fracture must he expected to occur when reaches a criticalvalue . The critical may be expected to be a material parameter.If is a material parameter the same value should be found by testing anotherspecimen of the same material but with a different size of the crack. Within certain

limits this is indeed the case. On the basis of this value the fracture strength ofcracks of any size in the same material can he predicted. It can also be predicted whichsize of crack can be tolerated in the material if stressed to a given level.

is a measure for the crack resistance of a material. Therefore is called the "planestrain fracture toughness". Materials with low fracture toughness can tolerate only small

cracks.

According to the elastic stress field solutions discussed in the previous which that the

stresses become infinite at the crack tip where r


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Fig. (3.4) stress distribution at the crack tip due to plastic zone

Irwin has argued that the occurrence of plasticity makes the crack behave as if it were

longer than its physical size. As a result of crack tip plasticity the displacements are

larger and the stiffness is lower than in the elastic case. In other words, the plate

behaves as if it contained a crack of somewhat larger size. The effective crack size,

, is equal to + . the physical crack size plus a correction . The quantity isknown as Irwins plastic zone correction. Assuming for the time being that the plastic

zone has a circular shape, the situation can be represented as in figure (3.4). where the

effective crack extends to the center of the plastic zone. If the plastic zone correction

is applied consistently a correction to K is also necessary.

= Plane stress 3.5

= 6

Plane strain 3.6

3.2 The crack opening displacement criterion (COD)

High strength materials usually have a low fracture toughness. Plane strain fracture

problems in these materials can be successfully treated by means of the fracture

mechanics procedures described in the two foregoing sections. These procedures are

known as the linear elastic fracture mechanics (LEFM) concepts. Since they are based

on elastic stress field equation The latter can be used if the size of the crack tip plastic

zone is small compared to the size of the crack. According to eq. (3.5) the plastic zone

size is proportional to /low strength. low yield strength materials usually have a


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high toughness. This means that the size of the plastic zone at fracture ( = ) maybe so large as compared to the crack size that LEFM do not apply. The latter is the case

if approaches unity. (The size of the plastic zone is also proportional to .At present, a versatile method to treat crack problems in high toughness materials is not

yet available. Wells has introduced the crack opening displacement (COD) concept for

such materials. Supposedly. Crack extension can take place when the material at the

crack tip has reached a maximum permissible plastic strain.

Crack extension or fracture is assumed to occur as soon as the crack opening

displacement exceeds a critical value. It can easily be equivalent this criterion to the

and criterion in the case where LEFM apply. This gives some confidence forthe supposed general validity. In the present stage of development, one of the

drawbacks of the COD criterion is the fact that it does not permit direct calculation of

a fracture stress. The critical COD for high toughness. Low strength materials isprimarily a comparative toughness parameter.

Dugdale also considers an effective crack which is longer than the physical crack as in

figure 3.5. The crack edge p. in front of the physical crack carry the yield stress. Tending

to close the crack. (The partpis not really cracked; the material can still bear the yield

stress). The size of p is chosen such that the stress singularity disappears.Kshould be

zero. This means that the stress intensity due to the uniform stress has to becompensated by the stress intensity, due to the wedge forces:

=

Figure( 3.5) Dugdale approach

a.

Dugdale crack: b. Wedge forces

Wells criterion is not in contradiction with LEFM. In the case of LEFM the elastic

solution for the crack opening displacement (COD) can still be used. The displacement

of the crack surfaces (figure 3.6)


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Fig. 3.6 crack opening displacement

= 2 = 4 3.7By applying a plastic zone correction, it follows that

= 4 ( + ) 3.8

Where + is the effective crack size and where the origin of the coordinatesystem is at the centre of the crack. The crack tip opening displacement at the tip of

the physical crack is found for x=a. Since

It turns out that:

= 4 2 3.9A displacement of the origin of the coordinate system to the crack tip yields the

general expression for crack opening:

= 4 2 3.10

CTOD then follows from = and leading to eq. 3.9Substitution of = 2 yields:

= 4

3.11

Eq (3.11) holds in the area of LEFM: fracture occurs if = which according toeq (3.11) is at a constant value of CTOD and it appears that Wells criterion applies in

LEFM.

Use of the criterion in LEFM would require measurement of CTOD. A direct

measurement of CTOD is difficult and virtual1y impossible in a routine test. It can be

obtained indirectly by measuring K and using eq. (3.11). That would imply acceptation


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of the factor 4/introduced by the plastic zone correction. The critical crack tip opening

displacement has been related to the critical values of fracture toughness.In general:

= () 3.12Where the factor (1 - ) can be deleted In plane stress situations and is a constantconstraint factor which theoretical analyses have shown to be in the range 12 and

which experimental measurement have shown to be approximately unity for both plane

stress and plane strain situations.

3.3 The -integralSo far, the discussions were limited to the case of linear elastic behavior with essentially

no crack tip plasticity. if there is appreciable plasticity. cannot be determined fromthe elastic stress field, since

may be affected considerably by the crack tip plastic

zone . Solutions for elastic-plastic behavior are not available, however, within certain

limitations; the -integral provides a means to determine an energy release rate forcases where plasticity effects are not negligible.

Fig. 3.7 crack tip coordinate system and arbitrary line integral contour

Eshelby has defined a number of contour integrals which are path independent by

virtue of the theorem of energy conservation. The two-dimensional form of one of

these integrals can be written as:

= 3.13With

= , = =

3.14


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Where

p=is a closed contour path surrounding the crack tip followed counter clockwise in a

stressed solid, W = is the strain energy per Unit volume, T= is the tension vector

(traction) perpendicular to nin the outside direction, u=is the displacement in the x-

direction, and s=is an element ofp(arc length).

From a more physical viewpointmay be interpreted as the potential energy differencebetween two identically loaded bodies having crack size () and (+). In this context

= 3.15Where

=material width and =strain energy or work done (area under the load-displacement curve ).

Naturally, in the linear elastic case = and therefore also = .Thus, one can postulate that crack growth or fracture occurs if exceeds a criticalvalue, which is analogous to, . and equal to if the material is essentiallylinear elastic. Hence, if one would accept the limitations, would be a fracture criterionapplicable to linear elastic as well as to plastic fracture. Measurement of in theelastic case is simple, because of its relationship to and . So, whether or not can be used as a more general criterion depends upon whether can be measuredeasily for a material that shows appreciable plasticity. In addition, of course, the

limitations of non-linear elasticity to deal with crack plasticity should be acceptable.But this object out of interesting.

The following equation are frequently used to relate the various fracture toughness

parameters:

= = = 3.16

Where

= for plane stress

= for plane strain

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